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Theorem dcomex 9858
Description: The Axiom of Dependent Choice implies Infinity, the way we have stated it. Thus, we have Inf+AC implies DC and DC implies Inf, but AC does not imply Inf. (Contributed by Mario Carneiro, 25-Jan-2013.)
Assertion
Ref Expression
dcomex ω ∈ V

Proof of Theorem dcomex
Dummy variables 𝑡 𝑠 𝑥 𝑓 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 1n0 8102 . . . . . . 7 1o ≠ ∅
2 df-br 5031 . . . . . . . 8 ((𝑓𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛) ↔ ⟨(𝑓𝑛), (𝑓‘suc 𝑛)⟩ ∈ {⟨1o, 1o⟩})
3 elsni 4542 . . . . . . . . 9 (⟨(𝑓𝑛), (𝑓‘suc 𝑛)⟩ ∈ {⟨1o, 1o⟩} → ⟨(𝑓𝑛), (𝑓‘suc 𝑛)⟩ = ⟨1o, 1o⟩)
4 fvex 6658 . . . . . . . . . 10 (𝑓𝑛) ∈ V
5 fvex 6658 . . . . . . . . . 10 (𝑓‘suc 𝑛) ∈ V
64, 5opth1 5332 . . . . . . . . 9 (⟨(𝑓𝑛), (𝑓‘suc 𝑛)⟩ = ⟨1o, 1o⟩ → (𝑓𝑛) = 1o)
73, 6syl 17 . . . . . . . 8 (⟨(𝑓𝑛), (𝑓‘suc 𝑛)⟩ ∈ {⟨1o, 1o⟩} → (𝑓𝑛) = 1o)
82, 7sylbi 220 . . . . . . 7 ((𝑓𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛) → (𝑓𝑛) = 1o)
9 tz6.12i 6671 . . . . . . 7 (1o ≠ ∅ → ((𝑓𝑛) = 1o𝑛𝑓1o))
101, 8, 9mpsyl 68 . . . . . 6 ((𝑓𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛) → 𝑛𝑓1o)
11 vex 3444 . . . . . . 7 𝑛 ∈ V
12 1oex 8093 . . . . . . 7 1o ∈ V
1311, 12breldm 5741 . . . . . 6 (𝑛𝑓1o𝑛 ∈ dom 𝑓)
1410, 13syl 17 . . . . 5 ((𝑓𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛) → 𝑛 ∈ dom 𝑓)
1514ralimi 3128 . . . 4 (∀𝑛 ∈ ω (𝑓𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛) → ∀𝑛 ∈ ω 𝑛 ∈ dom 𝑓)
16 dfss3 3903 . . . 4 (ω ⊆ dom 𝑓 ↔ ∀𝑛 ∈ ω 𝑛 ∈ dom 𝑓)
1715, 16sylibr 237 . . 3 (∀𝑛 ∈ ω (𝑓𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛) → ω ⊆ dom 𝑓)
18 vex 3444 . . . . 5 𝑓 ∈ V
1918dmex 7598 . . . 4 dom 𝑓 ∈ V
2019ssex 5189 . . 3 (ω ⊆ dom 𝑓 → ω ∈ V)
2117, 20syl 17 . 2 (∀𝑛 ∈ ω (𝑓𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛) → ω ∈ V)
22 snex 5297 . . 3 {⟨1o, 1o⟩} ∈ V
2312, 12fvsn 6920 . . . . . . . 8 ({⟨1o, 1o⟩}‘1o) = 1o
2412, 12funsn 6377 . . . . . . . . 9 Fun {⟨1o, 1o⟩}
2512snid 4561 . . . . . . . . . 10 1o ∈ {1o}
2612dmsnop 6040 . . . . . . . . . 10 dom {⟨1o, 1o⟩} = {1o}
2725, 26eleqtrri 2889 . . . . . . . . 9 1o ∈ dom {⟨1o, 1o⟩}
28 funbrfvb 6695 . . . . . . . . 9 ((Fun {⟨1o, 1o⟩} ∧ 1o ∈ dom {⟨1o, 1o⟩}) → (({⟨1o, 1o⟩}‘1o) = 1o ↔ 1o{⟨1o, 1o⟩}1o))
2924, 27, 28mp2an 691 . . . . . . . 8 (({⟨1o, 1o⟩}‘1o) = 1o ↔ 1o{⟨1o, 1o⟩}1o)
3023, 29mpbi 233 . . . . . . 7 1o{⟨1o, 1o⟩}1o
31 breq12 5035 . . . . . . . 8 ((𝑠 = 1o𝑡 = 1o) → (𝑠{⟨1o, 1o⟩}𝑡 ↔ 1o{⟨1o, 1o⟩}1o))
3212, 12, 31spc2ev 3556 . . . . . . 7 (1o{⟨1o, 1o⟩}1o → ∃𝑠𝑡 𝑠{⟨1o, 1o⟩}𝑡)
3330, 32ax-mp 5 . . . . . 6 𝑠𝑡 𝑠{⟨1o, 1o⟩}𝑡
34 breq 5032 . . . . . . 7 (𝑥 = {⟨1o, 1o⟩} → (𝑠𝑥𝑡𝑠{⟨1o, 1o⟩}𝑡))
35342exbidv 1925 . . . . . 6 (𝑥 = {⟨1o, 1o⟩} → (∃𝑠𝑡 𝑠𝑥𝑡 ↔ ∃𝑠𝑡 𝑠{⟨1o, 1o⟩}𝑡))
3633, 35mpbiri 261 . . . . 5 (𝑥 = {⟨1o, 1o⟩} → ∃𝑠𝑡 𝑠𝑥𝑡)
37 ssid 3937 . . . . . . 7 {1o} ⊆ {1o}
3812rnsnop 6048 . . . . . . 7 ran {⟨1o, 1o⟩} = {1o}
3937, 38, 263sstr4i 3958 . . . . . 6 ran {⟨1o, 1o⟩} ⊆ dom {⟨1o, 1o⟩}
40 rneq 5770 . . . . . . 7 (𝑥 = {⟨1o, 1o⟩} → ran 𝑥 = ran {⟨1o, 1o⟩})
41 dmeq 5736 . . . . . . 7 (𝑥 = {⟨1o, 1o⟩} → dom 𝑥 = dom {⟨1o, 1o⟩})
4240, 41sseq12d 3948 . . . . . 6 (𝑥 = {⟨1o, 1o⟩} → (ran 𝑥 ⊆ dom 𝑥 ↔ ran {⟨1o, 1o⟩} ⊆ dom {⟨1o, 1o⟩}))
4339, 42mpbiri 261 . . . . 5 (𝑥 = {⟨1o, 1o⟩} → ran 𝑥 ⊆ dom 𝑥)
44 pm5.5 365 . . . . 5 ((∃𝑠𝑡 𝑠𝑥𝑡 ∧ ran 𝑥 ⊆ dom 𝑥) → (((∃𝑠𝑡 𝑠𝑥𝑡 ∧ ran 𝑥 ⊆ dom 𝑥) → ∃𝑓𝑛 ∈ ω (𝑓𝑛)𝑥(𝑓‘suc 𝑛)) ↔ ∃𝑓𝑛 ∈ ω (𝑓𝑛)𝑥(𝑓‘suc 𝑛)))
4536, 43, 44syl2anc 587 . . . 4 (𝑥 = {⟨1o, 1o⟩} → (((∃𝑠𝑡 𝑠𝑥𝑡 ∧ ran 𝑥 ⊆ dom 𝑥) → ∃𝑓𝑛 ∈ ω (𝑓𝑛)𝑥(𝑓‘suc 𝑛)) ↔ ∃𝑓𝑛 ∈ ω (𝑓𝑛)𝑥(𝑓‘suc 𝑛)))
46 breq 5032 . . . . . 6 (𝑥 = {⟨1o, 1o⟩} → ((𝑓𝑛)𝑥(𝑓‘suc 𝑛) ↔ (𝑓𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛)))
4746ralbidv 3162 . . . . 5 (𝑥 = {⟨1o, 1o⟩} → (∀𝑛 ∈ ω (𝑓𝑛)𝑥(𝑓‘suc 𝑛) ↔ ∀𝑛 ∈ ω (𝑓𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛)))
4847exbidv 1922 . . . 4 (𝑥 = {⟨1o, 1o⟩} → (∃𝑓𝑛 ∈ ω (𝑓𝑛)𝑥(𝑓‘suc 𝑛) ↔ ∃𝑓𝑛 ∈ ω (𝑓𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛)))
4945, 48bitrd 282 . . 3 (𝑥 = {⟨1o, 1o⟩} → (((∃𝑠𝑡 𝑠𝑥𝑡 ∧ ran 𝑥 ⊆ dom 𝑥) → ∃𝑓𝑛 ∈ ω (𝑓𝑛)𝑥(𝑓‘suc 𝑛)) ↔ ∃𝑓𝑛 ∈ ω (𝑓𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛)))
50 ax-dc 9857 . . 3 ((∃𝑠𝑡 𝑠𝑥𝑡 ∧ ran 𝑥 ⊆ dom 𝑥) → ∃𝑓𝑛 ∈ ω (𝑓𝑛)𝑥(𝑓‘suc 𝑛))
5122, 49, 50vtocl 3507 . 2 𝑓𝑛 ∈ ω (𝑓𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛)
5221, 51exlimiiv 1932 1 ω ∈ V
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wex 1781  wcel 2111  wne 2987  wral 3106  Vcvv 3441  wss 3881  c0 4243  {csn 4525  cop 4531   class class class wbr 5030  dom cdm 5519  ran crn 5520  suc csuc 6161  Fun wfun 6318  cfv 6324  ωcom 7560  1oc1o 8078
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295  ax-un 7441  ax-dc 9857
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-ord 6162  df-on 6163  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-fv 6332  df-1o 8085
This theorem is referenced by:  axdc2lem  9859  axdc3lem  9861  axdc4lem  9866  axcclem  9868
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